The first thing that we should do, is draw a picture of the two parallelograms that we have . . . .
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Note: I put the letters of the angles inside the parallelograms, to save space.
There are a couple of things that you need to first notice: when a problem says that figures are similar, that means that they are the same SHAPE (which means that they will have the same angles), but it usually also means that they will be different SIZES. We know that PQRS is bigger than ABCD (in fact - if the ratio is 4:1, then that means that PQRS is 4 TIMES AS LARGE AS ABCD). Now that we know this, let's move onto our next step . . . .

Okay, now because we are given that the length of ABCD is 1/4 the length of PQRS, all we need to do, is find the actual length of side AB and multiply that by 4, to get the length of PQ. Now, in order to find the length of side AB, we need to use the distance formula, which is: \[d = \sqrt{(x-x)^2 + (y-y)^2}\] We need to take the coordinates of A and B, and insert them into the formula: \[d = \sqrt{(8-4)^2 + (6-6)^2}\] Then, we need to solve for "d". First, we subtract 4 from 8, to give: 4, then we square 4, which gives us: 16. Next, we need to subtract 6 from 6, to give: 0, then we square 0, which gives us: 0. Then, we need to add 16 to 0, to get: 16. Finally we take the square root of 16, which leaves us with a final answer of: d = 4 Then for the next step . . . . .

Now that we know that AB is 4 units long, we can figure out the length of PQ. If we think about it - PQ is 4 times as long as AB, so if we do 4 times the length of AB, then that will give us the length of PQ. So: 4 x 4 = 16. Yay! We now know that PQ is 16 units long. However, before we jump out of our chairs with joy :), we have a few more steps to take . . .

Okay, now we have to break this up into steps. Notice that P is located at: (-8,-5) and that Q is 16 units away from P (remember that the length of - or distance between - P and Q is 16). So, what we need to do, is add 16 to the coordinates of (-8,-5), to find the coordinates of Q. So, here's what we do . . . .

Okay, now we have to break this up into steps. Notice that P is located at: (-8,-5) and that Q is 16 units away from P (remember that the length of - or distance between - P and Q is 16). So, what we need to do, is add 16 to the coordinates of (-8,-5), to find the coordinates of Q. So, here's what we do . . . .

Ashley10116 - I'm so sorry this is taking so long. I'm trying to get some help for you, from one of the people who helps me. I'm a little stuck on this last step, so I'll let them help you with the rest of the problem. Sorry about that!!!!

Hey, Ashley10116 - a very kind friend - Calcmathlete - helped me with the answer. To make it faster, I'll just put what they said: "Alright. The line is indeed horizontal. However, it doesn't mean that it has to go along the x-axis. This means that you just have to use logic to figure out that the y-value of P stays the same in Q. THe only thing that changes is the x-value. SInce there is no reflection or translation in the new figure, you just have to add the distance, 16, to the new point. P(-8, -5) Q has the same y-value and in the original shape, B is the direct right of A. Just add 16. P(-8, -5) Q(8, -5)." In other words - the answer is the first choice (8,-5). Does this make sense?

Oh, I should explain to you that you first have to find the slope of the line, using A and B's coordinates and the formula (y-y)/(x-x) - when I plugged A and B's coordinates in, I got: (6-6)/(8-4) Which equaled: 0 So, because the slope turned out to be 0, that means that AB and PQ are both horizontal lines - that's what Calcmathlete was meaning when it says that the line is horizontal and there is no change in the y-axis (no change in the up or down movement of the line). Let me know if you need anything else to be explained. :)